What Are the Conditions for an Inflection Point?

[jd12345]

Well i know that an inflection point is where the curve changes its concavity.
But i don't really understand the conditions for it.
It says that second derivative should be zero(but that's not sufficient). I understand this. Second derivative being zero is not sufficient, example is y =x^4. So further condition is that some of the following odd derivatives should be zero which i don't understand

What's with the odd derivative? I don't get the intuition

 

[mathwonk]

concavity is measured buy the sign of the second derivative, so changing concavity means the second derivative changes sign.

If the second derivative changes sign at c, then in particular it has to be zero at c (if it exists there), but not vice versa. But if the third derivative is non zero at c, then the second derivative was either increasing or decreasing at c. Thus if the second derivative was zero at c and also the third derivative was not zero, then the second derivative must have changed sign...the same game goes on...A simple illustration is to think of the Taylor series. Your curve looks locally like the lowest non zero term of the Taylor series. So If the 5th derivative is the first non zero derivative then your curve looks like y = x^5, which has an inflection point,

but if the first non zero derivative is the 6th, then your curve looks like y = x^6, which has no inflection point.